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Total positivity and accurate computations related to $q$-Abel polynomials

Y. Khiar, E. Mainar, E. Royo-Amondarain, B. Rubio

Abstract

The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of $q$-calculus has been steadily growing in the literature. In this work the $q$-analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and $q$-Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of $q$-Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.

Total positivity and accurate computations related to $q$-Abel polynomials

Abstract

The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of -calculus has been steadily growing in the literature. In this work the -analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and -Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of -Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.
Paper Structure (6 sections, 9 theorems, 73 equations, 1 figure, 5 tables)

This paper contains 6 sections, 9 theorems, 73 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Notations and auxiliary results
  3. Factorization of matrix conversion between $q$-Abel and monomial bases
  4. Computations with $q$-Abel polynomials to HRA
  5. Numerical experiments
  6. Conclusions and final remarks

Key Result

Theorem 1

A given nonsingular matrix A is STP (resp., TP) if and only if the NE of $A$ and $A^T$ can be performed without row exchanges, all the multipliers of the NE of $A$ and $A^T$ are positive (resp., nonnegative), and the diagonal pivots of the Neville elimination of $A$ are all positive.

Figures (1)

  • Figure 1: The 2-norm conditioning of collocation matrices $A^{(q,\alpha)}$\ref{['eq:change']} with equidistant nodes $t_i=i/(n+1)$ for $i=1,\dots,n+1$, Wronskian matrices $W^{(q,\alpha)}$\ref{['eq:changeW']} for $x=50$ and Gramian matrices $G^{(q,\alpha)}$\ref{['eq:GramqAbel']} of $q$-Abel bases.

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • Theorem 3
  • ...and 4 more